Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (5-3*x^2-2*x)/(3+x+x^2)
-
Limit of (1+4*x)^(1/x)
-
Limit of (1+3/x)^(3*x)
-
Limit of (4+x^2)/(-6+2*x)
-
Identical expressions
- (one + four *x)^(one /x)
- (1 plus 4 multiply by x) to the power of (1 divide by x)
- (one plus four multiply by x) to the power of (one divide by x)
- (1+4*x)(1/x)
- 1+4*x1/x
- (1+4x)^(1/x)
- (1+4x)(1/x)
- 1+4x1/x
- 1+4x^1/x
- (1+4*x)^(1 divide by x)
-
Similar expressions
- (1-4*x)^(1/x)
Limit of the function (1+4*x)^(1/x)
The solution
You have entered
[src]
x _________ lim \/ 1 + 4*x x->0+
$$\lim_{x \to 0^+} \left(4 x + 1\right)^{\frac{1}{x}}$$
Limit((1 + 4*x)^(1/x), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+} \left(4 x + 1\right)^{\frac{1}{x}}$$
transform
do replacement
$$u = \frac{1}{4 x}$$
then
$$\lim_{x \to 0^+} \left(1 + \frac{4}{\frac{1}{x}}\right)^{\frac{1}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{4 u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{4 u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{4}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{4} = e^{4}$$
The final answer:
$$\lim_{x \to 0^+} \left(4 x + 1\right)^{\frac{1}{x}} = e^{4}$$
$$\lim_{x \to 0^+} \left(4 x + 1\right)^{\frac{1}{x}}$$
transform
do replacement
$$u = \frac{1}{4 x}$$
then
$$\lim_{x \to 0^+} \left(1 + \frac{4}{\frac{1}{x}}\right)^{\frac{1}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{4 u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{4 u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{4}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{4} = e^{4}$$
The final answer:
$$\lim_{x \to 0^+} \left(4 x + 1\right)^{\frac{1}{x}} = e^{4}$$
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
One‐sided limits
[src]
x _________ lim \/ 1 + 4*x x->0+
$$\lim_{x \to 0^+} \left(4 x + 1\right)^{\frac{1}{x}}$$
4 e
$$e^{4}$$
= 54.5981500331442
x _________ lim \/ 1 + 4*x x->0-
$$\lim_{x \to 0^-} \left(4 x + 1\right)^{\frac{1}{x}}$$
4 e
$$e^{4}$$
exp(4)
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-} \left(4 x + 1\right)^{\frac{1}{x}} = e^{4}$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(4 x + 1\right)^{\frac{1}{x}} = e^{4}$$
$$\lim_{x \to \infty} \left(4 x + 1\right)^{\frac{1}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(4 x + 1\right)^{\frac{1}{x}} = 5$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(4 x + 1\right)^{\frac{1}{x}} = 5$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(4 x + 1\right)^{\frac{1}{x}} = 1$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+} \left(4 x + 1\right)^{\frac{1}{x}} = e^{4}$$
$$\lim_{x \to \infty} \left(4 x + 1\right)^{\frac{1}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(4 x + 1\right)^{\frac{1}{x}} = 5$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(4 x + 1\right)^{\frac{1}{x}} = 5$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(4 x + 1\right)^{\frac{1}{x}} = 1$$
More at x→-oo
The graph